Arithmetic and geometric sequences and their summations Examples

Examples

The area of square \({A_1}{B_1}{C_1}{D_1}\) is \(1{\rm{ }}728{\rm{ c}}{{\rm{m}}^2}\).
The mid-points of the sides of square \({A_1}{B_1}{C_1}{D_1}\) are joined to form square \({A_2}{B_2}{C_2}{D_2}\) and the subsequent squares are also obtained by this method.
(See the figure)

(a) (i) Find the ratio of the area of square\({A_1}{B_1}{C_1}{D_1}\) to that of square \({A_2}{B_2}{C_2}{D_2}\).
(ii) Express the area of square \({A_n}{B_n}{C_n}{D_n}\) in terms of n.
(b) Find the area of square \({A_{10}}{B_{10}}{C_{10}}{D_{10}}\).

The sum of the 2nd and 3rd terms of a geometric sequence is 36, and the 4th term is 4 times the 2nd term.
(a) Find the common ratio of the sequence.
(b) Find the general term of the sequence.
(c) It is given that the common ratio of the sequence is negative. If all negative terms are picked to form a new sequence,
(i) find the general term of the new sequence.
(ii) determine whether the new sequence is a geometric sequence.

Solutions

(a) Let \({T_n} = a{r^{n - 1}}\) be the general term of the sequence,
where a and r are the first term and the common ratio of the sequence respectively.
∵ \({T_4} = 4{T_2}\)
∴ \(\begin{array}{1}a{r^{4 - 1}} = 4(a{r^{2 - 1}})\\a{r^3} = 4ar\\{r^2} = 4\end{array}\)
\(r = 2\) or -2